Stochastic Galerkin Methods without Uniform Ellipticity Marcus - - PowerPoint PPT Presentation

Stochastic Galerkin Methods without Uniform Ellipticity

Marcus Sarkis (WPI/IMPA) Collaborator: Juan Galvis (Texas A & M)

WPI/IMPA

RICAM MS & AEE-Workshop4, Dec13/2011

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 1 / 28

Problem of interest

Consider the Darcy’s equation    −∇x. (κ(x, ω)∇xu(x, ω)) = f (x, ω), for x ∈ D ⊂ Rd u(x, ω) = 0, on ∂D κ(x, ω) = eW (x,ω)

◮ W (x, ω) = ∞

k=1 ak(x)ξk(ω)

◮ ξk are iid standard normal random variables ◮ eW (x,ω) ∈ (0, ∞) not bounded, not uniformly elliptic

f (x, ω)

◮ Random forcing term Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 2 / 28

Outline

One-dimensional log-normal noise White noise framework Countable infinite-dimensional log-normal noise Galerkin spectral method Discretization, well-posedness, a priori error Numerical results Conclusions

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 3 / 28

Breeding analysis on Log-Normal without ellipticity

November 2005: I. Babuska, F. Nobile and R. Tempone: A stochastic collocation method for elliptic partial differential equations with random input data. March 2008: J. Galvis and S., Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. March 2009: X. Wan, B. Rozovskii and G. E. Karniadakis, A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. May 2009: C.J. Gittelson, Stochastic Galerkin discretization of the lognormal isotropic diffusion problem. June 2010: J. Charrier, Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. January 2011: A. Mugler and H.-J. Starkloff, On elliptic partial differential equations with random coefficients.

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 4 / 28

References

Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity, Juan Galvis and Marcus Sarkis. SIAM J.

• Numer. Anal., Vol. 47(5), pp. 3624-3651, 2009.

Regularity results for the ordinary product stochastic pressure equation, Juan Galvis and Marcus Sarkis. Submitted. Preprint serie IMPA A 692, 2011. An introduction to infinite-dimensional analysis, Giuseppe Da Prato. Universitext, Springer-Verlag, Berlin, 2006. Stochastic analysis, Ichiro Shigekawa. Translations of Mathematical Monographs, Vol. 224, AMS, Providence, RI, 2004.

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 5 / 28

A simple example: one-dimensional log-normal noise

−(ea1(x)ξ1ux)x = f (x), in D = (0, 1) Young Inequality: Let C1 := maxx∈D |a1(x)| and ǫ > 0 e−

C2 1 2ǫ e− ǫ 2 ξ2 1 ≤ ea1(x)ξ1 ≤ e C2 1 2ǫ e ǫ 2 ξ2 1

Idea: Use weights of the type esξ2

1 (easy to integrate)

• G(ξ1)esξ2

1dξ1 =

1 √ 2π ∞

−∞

G(y)esy2− 1

2 y2dy

Lax Milgram: ux(·, ξ1)2

L2(0,1) ≤ C 2 Pe

C2 1 ǫ f 2

H−1(0,1)eǫξ2

1 Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 6 / 28

A simple example: one-dimensional log-normal noise

ux(·, ξ1)2

L2(D) ≤ C 2 Pe

C2 1 ǫ f 2

H−1(D)eǫξ2

1

Integrate over ξ1, (0 < ǫ < 1/2) u ∈ H1

0(D) ⊗ (L2)(dξ1) bounded by f ∈ H−1(D) ⊗ (L2)ǫ(dξ1)

Integrate over ξ1, (ǫ > 0 and s ∈ R such that s + ǫ < 1/2)

• ||ux(·, ξ1)||2

L2(D)esξ2

1dξ1

= 1 √ 2π ∞

−∞

ux(·, y)2

L2(D)esy2− 1

2 y2dy

≤ C 2

Pe

C2 1 ǫ f 2

H−1

1 √ 2π ∞

−∞

e(s+ǫ)y2− 1

2 y2dy

= C 2

Pe

C2 1 ǫ (1 − 2(s + ǫ))− 1 2 f 2

H−1(D)

For s = 0, ux ∈ L2(D) ⊗ L2

s(dξ1) important when f (x, ξ1)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 7 / 28

A simple example: one-dimensional log-normal noise

RHS f (x, ξ1) with lack (s < 0) extra (s > 0 ) decay in ξ1

• ux(·, ξ1)2

L2(D)esξ2

1dξ1 ≤ C 2

Pe

C2 1 ǫ

• f (·, ξ1)2

H−1(D)e(s+ǫ)ξ2

1dξ1

Note: u → s, f → s + ǫ, and v → −(s + ǫ) Given f ∈ H−1(D) ⊗ L2

s+ǫ(dξ1), find u ∈ H1 0(D) ⊗ L2 s(dξ1) such that

a(u, v) = f (v), ∀v ∈ H1

0(D) ⊗ L2 −(s+ǫ)(dξ1)

Existence and uniqueness (inf-sup condition) Galvis and S. (09’)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 8 / 28

We need a theoretical framework to deal:

Infinite-dimensional case (Gaussian measure) Generalized Wiener-chaos expansions (on Hilbert spaces) Galerkin spectral methods (Explicit computations) A priori error estimates (Natural to establish) Regularity theory (Derivatives in ω and x) Gaussian Sobolev (Hilbert) spaces Constants that depend on few quantities

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 9 / 28

Probability space: constructed from a pair (H, A)

Using KL:

◮ H := L2(D) ◮ Correlation operator: C v(x) :=

• D k(x, ˜

x)v(˜ x)d˜ x

◮ C →: Eigenfunctions qk, eigenvalues µk ◮ A = C −1 ◮ A →: Eigenfunctions qk, eigenvalues λk = 1/µk

Using convolution (1D- Smoothed white noise)

◮ H := L2(R) ◮ A := − d2

dx2 + x2 + 1

◮ A →: Hermite functions qk, eigenvalues λk = 2k Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 10 / 28

White noise framework

Hilbert space H, operator A, and H-orthonormal basis {qk}∞

k=1:

◮ Aqk = λkqk, k = 1, 2, . . . ◮ 1 < λ1 ≤ λ2 ≤ · · · ◮ ∞

k=1 λ−2θ k

< ∞ for some constant θ > 0

p ≥ 0, ξ ∈ H, ξ2

p := Apξ2 H = ∞ k=1 λ2p k (ξ, qk)2 H

Sp := {ξ ∈ H; ξp < ∞} S := ∩p≥0Sp S′ Probability measure µ (Bochner-Minlos theorem) characterized by Eµei·,ξ :=

• S′ eiω,ξdµ(ω) = e− 1

2 ξ2 H, for all ξ ∈ S

Probability space (Ω, F, P) = (S′, B(S′), µ)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 11 / 28

Remarks: Fernique and change of variables

µ(S−θ) = 1 where S−θ = {ω ∈ S′ : ω−θ < ∞} Eµei·,ξ :=

• S−θ

eiω,ξdµ(ω) = e− 1

2 ξ2 H for all ξ ∈ S

(S−θ, B(S−θ), µ): normally distributed RV Xξ(ω) = ω, ξ Equivalent formulation: E˜

µei·,ξ :=

• H

eih,ξd ˜ µ(h) = e− 1

2 ξ2 A−2θ for all ξ ∈ S

˜ µ Gaussian measure with covariance A−2θ (H, B(H), ˜ µ): normally distributed RV Yξ(h) = h, Aθξ See Da Prato (06’) for θ = 1/2 and Galvis and S. (11’)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 12 / 28

Gaussian Field W (x, ω) = ω, φx

In KL:

◮ φx(˜

x) ≡ φ(x, ˜ x) = ∞

k=1 λ − 1

2

k

qk(x)qk(˜ x)

◮ W (x, ω) = ω, φx = ∞

k=1 ak(x)ω, qk

◮ ak(x) = λ

− 1

2

k

qk(x)

◮ ξk(ω) = ω, qk i.i. normally distributed

In convolution:

◮ φx(˜

x) ≡ φ(x − ˜ x) (smoothed window)

◮ W (x, ω) = ω, φx = ∞

k=1 ak(x)ω, qk

◮ ak(x) =

• R φ(x − ˜

x)qk(˜ x)d˜ x

◮ ξk(ω) = ω, qk i.i. normally distributed Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 13 / 28

Countable independent normals

W (x, ω) = ω, φx = ∞

k=1 ak(x)ξk(ω)

−λ2θ

k a2 k(x)

2ǫ − ǫλ−2θ

k

ξk(ω)2 2 ≤ ak(x)ξk(ω) ≤ λ2θ

k a2 k(x)

2ǫ + ǫλ−2θ

k

ξk(ω)2 2 −|||φ|||2

θ

2ǫ − ǫω2

−θ

2 ≤ ω, φx ≤ |||φ|||2

θ

2ǫ + ǫω2

−θ

2 |||φ|||2

θ := supx∈D φx2 θ

φx2

θ = ∞

• k=1

λ2θ

k (φx, qk)2 H = ∞

• k=1

λ2θ

k a2 k(x) < ∞

ω2

−θ := ∞ k=1 λ−2θ k

ω, qk2

• S′ ω2

−θdµ(ω) = ∞

• k=1

λ−2θ

k

< ∞

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 14 / 28

Two examples

KL:

◮ ∞

k=1

√µk < ∞

◮ supk qkL∞(D) < ∞ ◮ Take θ = 1/4, both conditions are satisfied

Smoothed white noise:

◮ λk = 2k (note that ∞

k=1 λ−1 k

= ∞)

◮ φ(x − ˜

x) a smooth window (the ak(x) decay fast)

◮ Take θ > 1, both conditions are satisfied Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 15 / 28

Generalization to infinite-dimension

Ellipticity κmin(ω) := e−

|||φ|||2 θ 2ǫ

−

ǫω2 −θ 2

≤ eω,φx Lax-Milgram (fixed ω) |u(·, ω)|2

H1(D) ≤

C 2

P

κmin(ω)2 f (·, ω)2

H−1(D)

For ǫ > 0 and s ∈ ℜ |u(·, ω)|2

H1

0(D)esω2 −θ ≤ C 2

Pe

|||φ|||2 θ ǫ

f (·, ω)2

H−1(D)e(s+ǫ)ω2

−θ

Solution space and test space. Stability |u|2

H1(D)×(L2)s ≤ C 2 Pe

|||φ|||2 θ ǫ

f 2

H−1(D)×(L2)s+ǫ

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 16 / 28

Existence and uniqueness

Given f ∈ H−1 × (L2)s+ǫ Find u ∈ H1

0 × (L2)s such that for all v ∈ H1 0 × (L2)−s−ǫ

• S′×D

eω,φx∇u(x, ω)∇v(x, ω)dxdµ =

• S′×D

f (x, ω)v(x, ω)dxdµ Existence and uniqueness (inf-sup condition) Details in Galvis and S’ (09’)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 17 / 28

Generalized Hermite polynomials in (L)s norm

Multi-indices J : α = (α1, α2, . . . ) ∈ (NN

0 )c

Order of α : d(α) := max {k : αk = 0} Length of α : |α| := α1 + α2 + · · · + αd(α) Note esλ−2θ

k

y2− y2

2 = e

−

1 2σ2 k

y2

if σk(s) :=

• 1 − 2s

λ2θ

k

− 1

2

σ∗(s) :=

• S′ esω2

−θdµ(ω) =

∞

k=1 σk(s)

s < λ2θ

1

2

+∞ s ≥ λ2θ

1

2

σk-Hermite polynomials hσ2

k,αk, orthogonal in L2(R, e

−

1 2σ2 k

y2

dy) s < λ2θ

1

2 , α = (α1, α2, . . . ) ∈ J and σ(s) = (σ1(s), σ2(s), . . . ), define

Hσ2(s),α(ω) := 1

• σ∗(s)

d(α)

• k=1

hσ2

k(s),αk(ω, qk);

ω ∈ S′.

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 18 / 28

Wiener-chaos expansion in the Hm × (L)s norm

Wiener-chaos basis (for s < λ2θ

1

2 )

Hσ2(s),α2

(L)s = α!σ(s)2α

α! = α1!α2! · · · αd(α)! σ(s)2α = σ1(s)2α1σ2(s)2α2 · · · σd(α)(s)2αd(α) z ∈ (L)s represented by a Wiener-chaos expansion z =

• α∈J

zα,sHσ(s)2,α with z2

(L)s =

• α∈J

α!σ(s)2αz2

α,s

u ∈ Hm × (L)s u2

Hm×(L)s =

• α∈J

α!σ(s)2αuα,s2

Hm(D)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 19 / 28

Weighted chaos norms

z ∈ (L)s represented by a Wiener-Chaos expansion z =

• α∈J

zα,sHσ(s)2,α with z2

(L)s =

• α∈J

α!σ(s)2αz2

α,s

z ∈ (L)p;s and weighted chaos norms z2

p;s :=

• α∈J

(1 + α, λ2p) α!σ(s)2αz2

α,s,

α, λ = α1λ1 + α2λ2 + · · · + αd(α)λd(α) Measure how fast the chaos coefficients decay u ∈ Hm × (L)p;s u2

Hm×(L)p;s =

• α∈J

(1 + α, λ2p) α!σ(s)2αuα,s2

Hm(D)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 20 / 28

Analysis

Isomorphism between weighted chaos norms and Sobolev stochastic

• derivatives. Galvis and S. (11’)

Weighted chaos norms: easy for establishing a priori error estimates. Galvis and S. (09’) and (11’) Stochastic derivatives: easier for establishing regularity theory. Galvis and S. (11’)

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 21 / 28

Finite dimensional discretization

FEM spatial discretization X h

0 (D) ⊂ H1 0(D)

Let N, K ∈ N0 and define J N,K := {α ∈ J : d(α) ≤ K, and, |α| ≤ N} Polynomials in ω, q1, . . . , ω, qK of total degree at most N PN,K := span

• Hσ(s)2,α : α ∈ J N,K

QK is the (H-orthogonal) projection on the span{q1, . . . , qK} QKω :=

K

• k=1

ω, qkqk, for all ω ∈ S′.

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 22 / 28

Finite dimensional formulation

Solution space: X N,K,h := X h

0 (D) × PN,K

Test space: YN,K,h

s

:=

• v : v(x, ω) = ˜

v(x, ω)e(s+ ǫ

2 )PK ω2 −θ, ˜

v ∈ X N,K,h Find uN,K,h ∈ X N,K,h such that a(uN,K,h, v) = f , v for all v ∈ YN,K,h

s

Discrete inf-sup conditions, existence and uniqueness

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 23 / 28

A priori error estimate

U1

s := H1(D) × (L)s

Let ǫ > 0, s + 2ǫ < λ2θ

1

2

and −s − ǫ <

λ2θ

K+1

2 . Then

|u − uN,K,h|U1

s ≤

• 1 + e

|||φ|||2 θ ǫ

∞

• k=K+1

σk(−s − ǫ)

• inf

z∈X N,K,h |u − z|U1

s+2ǫ

infz∈X N,K,h |u − z|U1

s+2ǫ bounded by

max

• 1

1 + (N + 1)λ1 , 1 1 + λK+1 p |u|U1

p;s+2ǫ + ˆ

Ch|u|U2

s+2ǫ

U1

p;s+2ǫ := H1(D) × (L)p;s+2ǫ

U2

s+2ǫ := H2(D) × (L)s+2ǫ

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 24 / 28

Numerical experiments 1D

Smoothed white noise (convolution method) Window φ : R → R D = [0, 1] and φ(x) = e− 1

2 x2

T = − d2

dx2 + x2 + 1, Tqk = (2k)qk and λk = 2k

Hermite functions orthonormal in L2(R): qk(x) := 1 √π(k − 1)! e− 1

2 x2hk−1(

√ 2x), k = 1, 2, . . . ak(x) =

• R φ(x − ˜

x)qk(˜ x)d˜ x Exact solution u = x(1 − x) 2 e

∞

k=1 ak(x)yk Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 25 / 28

Numerical experiments

k 1 2 3 4 5 K+N

K

• 1

2 6 20 70 252 |u − QN,Ku|U1 1.6284 1.3761 0.9767 0.6162 0.3570 0.1920 (1.18) (1.41) (1.59) (1.73) (1.86) |u − uN,K,h|U1 1.7292 1.6157 1.3590 1.0375 0.7281 0.4626 1.7291 1.6153 1.3575 1.0340 0.7214 0.4659 (1.07) (1.18) (1.31) (1.43) (1.55) |u − uN,K,h|κ 0.4319 0.3691 0.2598 0.1573 0.0836 0.0454 0.4318 0.3688 0.2589 0.1552 0.0790 0.0279 (1.17) (1.42) (1.67) (1.96) (2.83) Errors for K = N = k, h = 1/16, 1/32 and ǫ = 1

2, s = 0. For

h = 1/32 we have added in parenthesis the reduction factor, when passing to next value of k, corresponding to the projection and finite element error in the seminorm | · |U1

0 and the finite element error in

the κ-energy norm.

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 26 / 28

Numerical experiments

Approximation of u(0,0,0,... ) for K = N = 3, h = 1

10 and ǫ = 0, 1, 2

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 27 / 28

Conclusions

Ellipticity treatment Unified framework for KL and smoothed white noise More general f and infinite-dimensional case Weighted norms, well-posedness, a priori error estimates Framework for establishing regularity theory

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 28 / 28